4.2.3.2      The Resistance (R) Curve Approach

If the crack tip plastic zone size is estimated to be on the order of the structural thickness but substantially smaller than other geometrical variables (crack length, ligament size, height, etc.), a linear elastic fracture mechanics analysis can still be sensibly used to predict the catastrophic cracking event.  The failure criterion for tearing type fractures under these conditions states that fracture will occur when (1) the stress-intensity factor K reaches or exceeds the material’s fracture resistance K_R and (2) the rate of change of K (with respect to crack length) reaches or exceeds the rate of change of K_R (with respect to crack length).  Physically, the criterion means that at failure, the energy available to extend the crack equals or exceeds the material resistance to crack growth.  The failure criterion becomes simply,

The corresponding applied stress, s_f, at this point is defined as the fracture stress that determines the residual strength of the cracked structure.  The criterion presented in Equation 4.2.4 is noted to be a two-parameter criterion rather than the single parameter criteria that was presented in paragraph 4.2.3.1.  To interpret the meaning of this criterion, first consider the structural parameters that are a function of the geometry and stress, i.e. K and ÐK/Ða.

In general, the estimation of K involves the relationship K = sbÖpa as given in Section 2; using this equation, the variation of K with respect to crack length (a) can be obtained for various values of stress (s) as shown in Figure 4.2.11a.  Shown in Figure 4.2.11b is the variation of K_R with respect to the crack extension (Da) that was developed for the given material using the procedures outlined in Figure 4.2.7.  Since this R-curve is assumed to be independent of the initial crack length, it can be superimposed on the plot of K versus a as shown in Figure 4.2.11c.  The tangency point between the applied stress intensity factor curve (K vs. a) and the R-curve (K_R vs. Da) determines the commencement of unstable crack propagation.  In general, the accurate method of determining the tangency point involves the numerical solution based on the experimentally obtained R-curve.  Using a least squares determined polynomial expression for R-curve and knowing an expression for K in terms of crack length, the common tangent point can be obtained by equating the functional values (K = K_R) and also the first derivatives with respect to the crack length dK/da = dK_R /da of these two expressions.

Figure 4.2.11.  Schematic Illustration of the Individual and Collective Parts of a K_R Fracture Analysis

The slow stable tear is dependent on a structural configuration in which the plastic zone at the crack tip is no longer negligible but not enormous.  Krafft, et al. [1961], Srawley & Brown [1965], and McCabe [1973] explain the dependence of the R-curve on structural configuration as well as with test procedures used to evaluate the R-curve.  See Section 7 for additional information on test procedures and the Damage Tolerant Design (Data) Handbook [Skinn, et al., 1994] for a summary of available data.